Calculus Examples

$f (x) = 3 x^{2} {(x^{3} + 1)}^{7}$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2} {(x^{3} + 1)}^{7}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2} {(x^{3} + 1)}^{7}]$ .

$3 \frac{d}{d x} [x^{2} {(x^{3} + 1)}^{7}]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = {(x^{3} + 1)}^{7}$ .

$3 (x^{2} \frac{d}{d x} [{(x^{3} + 1)}^{7}] + {(x^{3} + 1)}^{7} \frac{d}{d x} [x^{2}])$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{7}$ and $g (x) = x^{3} + 1$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $x^{3} + 1$ .

$3 (x^{2} (\frac{d}{d u} [u^{7}] \frac{d}{d x} [x^{3} + 1]) + {(x^{3} + 1)}^{7} \frac{d}{d x} [x^{2}])$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 7$ .

$3 (x^{2} (7 u^{6} \frac{d}{d x} [x^{3} + 1]) + {(x^{3} + 1)}^{7} \frac{d}{d x} [x^{2}])$

Step 3.3

Replace all occurrences of $u$ with $x^{3} + 1$ .

$3 (x^{2} (7 {(x^{3} + 1)}^{6} \frac{d}{d x} [x^{3} + 1]) + {(x^{3} + 1)}^{7} \frac{d}{d x} [x^{2}])$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $x^{3} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [1]$ .

$3 (x^{2} (7 {(x^{3} + 1)}^{6} (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [1])) + {(x^{3} + 1)}^{7} \frac{d}{d x} [x^{2}])$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 (x^{2} (7 {(x^{3} + 1)}^{6} (3 x^{2} + \frac{d}{d x} [1])) + {(x^{3} + 1)}^{7} \frac{d}{d x} [x^{2}])$

Step 4.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$3 (x^{2} (7 {(x^{3} + 1)}^{6} (3 x^{2} + 0)) + {(x^{3} + 1)}^{7} \frac{d}{d x} [x^{2}])$

Step 4.4

Simplify the expression.

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Step 4.4.1

Add $3 x^{2}$ and $0$ .

$3 (x^{2} (7 {(x^{3} + 1)}^{6} (3 x^{2})) + {(x^{3} + 1)}^{7} \frac{d}{d x} [x^{2}])$

Step 4.4.2

Multiply $3$ by $7$ .

$3 (x^{2} (21 {(x^{3} + 1)}^{6} x^{2}) + {(x^{3} + 1)}^{7} \frac{d}{d x} [x^{2}])$

Step 5

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 5.1

Move $x^{2}$ .

$3 (x^{2} x^{2} (21 {(x^{3} + 1)}^{6}) + {(x^{3} + 1)}^{7} \frac{d}{d x} [x^{2}])$

Step 5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 (x^{2 + 2} (21 {(x^{3} + 1)}^{6}) + {(x^{3} + 1)}^{7} \frac{d}{d x} [x^{2}])$

Step 5.3

Add $2$ and $2$ .

$3 (x^{4} (21 {(x^{3} + 1)}^{6}) + {(x^{3} + 1)}^{7} \frac{d}{d x} [x^{2}])$

Step 6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$3 (x^{4} (21 {(x^{3} + 1)}^{6}) + {(x^{3} + 1)}^{7} (2 x))$

Step 7

Move $2$ to the left of ${(x^{3} + 1)}^{7}$ .

$3 (x^{4} (21 {(x^{3} + 1)}^{6}) + 2 \cdot {(x^{3} + 1)}^{7} x)$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$3 (x^{4} (21 {(x^{3} + 1)}^{6})) + 3 (2 {(x^{3} + 1)}^{7} x)$

Step 8.2

Multiply $21$ by $3$ .

$63 (x^{4} ({(x^{3} + 1)}^{6})) + 3 (2 {(x^{3} + 1)}^{7} x)$

Step 8.3

Multiply $2$ by $3$ .

$63 x^{4} {(x^{3} + 1)}^{6} + 6 ({(x^{3} + 1)}^{7} x)$

Step 8.4

Factor $3 x {(x^{3} + 1)}^{6}$ out of $63 x^{4} {(x^{3} + 1)}^{6} + 6 {(x^{3} + 1)}^{7} x$ .

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Step 8.4.1

Factor $3 x {(x^{3} + 1)}^{6}$ out of $63 x^{4} {(x^{3} + 1)}^{6}$ .

$3 x {(x^{3} + 1)}^{6} (21 x^{3}) + 6 {(x^{3} + 1)}^{7} x$

Step 8.4.2

Factor $3 x {(x^{3} + 1)}^{6}$ out of $6 {(x^{3} + 1)}^{7} x$ .

$3 x {(x^{3} + 1)}^{6} (21 x^{3}) + 3 x {(x^{3} + 1)}^{6} (2 (x^{3} + 1))$

Step 8.4.3

Factor $3 x {(x^{3} + 1)}^{6}$ out of $3 x {(x^{3} + 1)}^{6} (21 x^{3}) + 3 x {(x^{3} + 1)}^{6} (2 (x^{3} + 1))$ .

$3 x {(x^{3} + 1)}^{6} (21 x^{3} + 2 (x^{3} + 1))$